Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas

We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass $8/(3\sqrt{\pi })$, giving the depletion fraction $\frac{8}{3\sqrt{\pi }}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$ for density ρ and hard-sphere exclusion distance a.